Supersymmetric Partner Damping of Free Damping

We determine the class of damped modes ỹ which are related to the common free damping modes y by supersymmetry. They are obtained by employing the factorization of Newton’s differential equation of motion for the free damped oscillator by means of the general solution of the corresponding Riccati equation together with Witten’s method of constructing the supersymmetric partner operator. This procedure introduces one-parameter families of (transient) modes for each of the three types of free damping, corresponding to a particular type of anti-restoring acceleration (adding up to the usual Hooke restoring acceleration) of the form a(t) = 2γ 2 (γt+1) ỹ, where γ is the family parameter that has been chosen as the inverse of the Riccati integration constant. In supersymmetric terms, they represent all those damping modes having the same free damping partner mode. PACS number(s): 11.30.Pb, 43.40At e-mail: [email protected] e-mail: [email protected] 1 The damped oscillator (DO) is a cornerstone of physics and a primary textbook example in classical mechanics. Schemes of analogies allow its extension to many areas of physics where the same basic concepts occur with merely a change in the meaning of the symbols. Apparently, there might hardly be anything new to say about such an obvious case. However, in the following we would like to exhibit a new and nice feature of damping resulting from the mathematical procedure of factorization of its differential equation. In the past, the factorization of the DO differential equation (Newton’s law) has been tackled by a few authors [1] but not in the framework that will be presented herein. Namely, recalling that such factorizations are common tools in Witten’s supersymmetric quantum mechanics [2] and imply particular solutions of Riccati equations known as superpotentials, we would like to explore here the factoring of the DO equation by means of the general solution of the Riccati equation, a procedure which has been first used in physics by Mielnik [3] for the quantum harmonic oscillator. In other words, we shall exploit the non-uniqueness of the factorization of second-order differential operators, on the example of the classical damped oscillator. We write the ordinary DO Newton’s law in the form Ny ≡ ( d2 dt2 + 2β d dt + β ) y = (β − ω 0)y = αy , (1) i.e., we already added a β2y term in both sides in order to perform the factoring. The coefficient 2β is the friction constant per unit mass and ω0 is the natural frequency of the oscillator. The factorization ( d dt + β )( d dt + β ) y = αy (2) follows, and previous authors [1] discussed the classical cases of underdamping (α2 < 0), critical damping (α2 = 0), and overdamping (α2 > 0) in terms of the first order differential equation Ly ≡ ( d dt + β )